I've always thought this was the least masochistic way to prove these identities because of the efficient notation, but this is notation the questioner likely hasn't encountered.
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David HMay 26 '14 at 8:18

@David H Yes, the other brute-force methods are not a good use of time in my estimation. With the index notation you can more clearly realize why these sort of identities are actually true!
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James S. CookMay 26 '14 at 8:20